"Just tabled amendment to Criminal Justice Bill to make life just a bit harder for cyber-bullies and sex pests using texts to harass victims," the politico said on her Twitter account.

According to the Evening Standard, Grayling agreed that the legislation needed to be tightened to protect victims from malicious comments being directed at them on free content ad networks such as Twitter and Facebook.

It comes in light of some high profile cases over the past year involving celebrities and others in the public eye who complained to the police, after being verbally attacked online with death threats and abusive messages.

Perhaps the most notorious example was the one involving feminist campaigner Caroline Criado-Perez. Two people were later found guilty of sending menacing communications after Criado-Perez made a formal statement to police in July 2013 about being harassed online.